#143153

order	512
nilpotency class	4
isoclinism	isoclinic class of #143148
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

416 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9)(4,12)(5,8)(6,10)(7,14)(11,16)(13,15)(17,18,25,32)(19,33,37,27)(20,26,40,46)(21,29,41,48)(22,28,38,42)(23,31,24,30)(34,45,39,44)(35,47,36,43);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14)(17,19,25,37)(18,26,32,46)(20,38,40,22)(21,39,41,34)(23,36,24,35)(27,29,33,48)(28,47,42,43)(30,45,31,44);(6,15)(9,16)(17,20,25,40)(18,27,32,33)(19,21,37,41)(22,36,38,35)(23,39,24,34)(26,42,46,28)(29,44,48,45)(30,43,31,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50349);